Notation: $M$ is a DFA; $L(M)$ is the language accepted by $M$; $\min(M)$ is the minimal automaton equivalent to $M$ derived from a minimization algorithm such as the Hopcroft algorithm; and $|M|$ is the size of $M$: the number of states in $M$.

We are given the alphabet $\{0,1\}$ and some $n \in \mathbb{N}$.

Let's define some sets to set up the question.

$$ A = \{M \mid L(M) \subseteq \{0,1\}^n\}$$

So $A$ is the set of all automata that accept some language whose words are composed of $n$-bit binary strings. My intention here is also to require that all members of $A$ reject strings of other lengths. Building from this, consider

$$ A_{\min} = \{ \min(M) \mid M \in A \} $$

So $A_{\min}$ is the set of all minimized automata from $A$. Building further, let

$$ x = \max \{ |M| \mid M \in A_{\min}\} $$

So $x$ is the size of the largest automaton from $A_{\min}$. Now we can specify the automaton or automata we're looking for:

$$S = \{ M \mid M \in A_{\min} \, \, \text{and} \, \, |M| = x \} $$

How would one go about constructing a member (or members) of $S$? Any will do, but simpler constructions are preferred.

My initial naive attempt to do this failed. I tried building it up by induction, starting from a basis state which had two states: an accepting state and a non-accepting state. For the inductive step, I tried to build a binary tree composed of two different subtrees built from the previous step. This failed because minimization still merged common subtrees together.

  • $\begingroup$ @D.W. Thanks for your ongoing help and patience with this question. I've substantially rewritten the question to make things more precise. To address your specific questions: "over" is formalized by set $A$. $A$ also addresses your first two questions. Your third question is addressed by set $S$. Your final question is addressed by $min(M)$ in the notation section. Please let me know what other improvements I can make so this question can be reopened. Many thanks $\endgroup$
    – ShyPerson
    May 19 '21 at 20:49
  • 1
    $\begingroup$ That helped a ton! Thank you. $\endgroup$
    – D.W.
    May 19 '21 at 20:55
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    $\begingroup$ Here's a construction that might get close to the right asymptotics: let $\Sigma=\{0,1\}^{\lg n}$; there's an isomorphism $\Sigma^{n/\lg n} \to \{0,1\}^n$; then consider the language of all words $w$ over $\Sigma^{n/\lg n}$ such that at least one $s \in \Sigma$ does not appear in $w$, i.e., $L=\{w \in \Sigma^{n/\lg n} \mid \exists s \in \Sigma . w \in (\Sigma \setminus \{s\})^{n/\lg n}\}$. Then by cs.stackexchange.com/q/3381/755, the minimal DFA for $L$ has at least $2^{n/\lg n}$ states. It's easy to see that $x \le n \cdot 2^n$ states. $\endgroup$
    – D.W.
    May 19 '21 at 22:15
  • $\begingroup$ @D.W. Many thanks again for all your help! $\endgroup$
    – ShyPerson
    May 20 '21 at 2:45

Your question is solved in Cezar Câmpeanu and Wing Hong Ho, The Maximum State Complexity for Finite Languages, Corollary 10.

  • $\begingroup$ Thanks! If I have follow-up questions, do I ask them here or in a new question? $\endgroup$
    – ShyPerson
    May 20 '21 at 3:18
  • $\begingroup$ Depends on the question. $\endgroup$ May 20 '21 at 3:46

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